Scilab Function
Last update : April 1993

cainv - Dual of abinv

### Calling Sequence

  `[X,dims,J,Y,k,Z]=cainv(Sl,alfa,beta,flag)`

### Parameters

- `Sl`: `syslin` list containing the matrices `[A,B,C,D]`.
- `alfa`: real number or vector (possibly complex, location of closed loop poles)
- `beta`: real number or vector (possibly complex, location of closed loop poles)
- `flag`: (optional) character string `'ge'` (default) or `'st'` or `'pp'`
- `X`: orthogonal matrix of size nx (dim of state space).
- `dims`: integer row vector `dims=[nd1,nu1,dimS,dimSg,dimN]` (5 entries, nondecreasing order).If `flag='st'`, (resp. `'pp'`), `dims` has 4 (resp. 3) components.
- `J`: real matrix (output injection)
- `Y`: orthogonal matrix of size ny (dim of output space).
- `k`: integer (normal rank of `Sl`)
- `Z`: non-singular linear system (`syslin` list)

### Description

`cainv` finds a bases `(X,Y)` (of state space and output space resp.) and output injection matrix `J` such that the matrices of Sl in bases (X,Y) are displayed as:

                      [A11,*,*,*,*,*]                [*]
                      [0,A22,*,*,*,*]                [*]
       X'*(A+J*C)*X = [0,0,A33,*,*,*]   X'*(B+J*D) = [*]
                      [0,0,0,A44,*,*]                [0]
                      [0,0,0,0,A55,*]                [0]
                      [0,0,0,0,0,A66]                [0]

              Y*C*X = [0,0,C13,*,*,*]          Y*D = [*]
                      [0,0,0,0,0,C26]                [0]

The partition of `X` is defined by the vector `dims=[nd1,nu1,dimS,dimSg,dimN]` and the partition of `Y` is determined by `k`.

Eigenvalues of `A11``(nd1 x nd1)` are unstable. Eigenvalues of `A22``(nu1-nd1 x nu1-nd1)` are stable.

The pair `(A33, C13)``(dimS-nu1 x dimS-nu1, k x dimS-nu1)` is observable, and eigenvalues of `A33` are set to `alfa`.

Matrix `A44``(dimSg-dimS x dimSg-dimS)` is unstable. Matrix `A55``(dimN-dimSg,dimN-dimSg)` is stable

The pair `(A66,C26)``(nx-dimN x nx-dimN)` is observable, and eigenvalues of `A66` set to `beta`.

The `dimS` first columns of `X` span S= smallest (C,A) invariant subspace which contains Im(B), `dimSg` first columns of `X` span Sg the maximal "complementary detectability subspace" of `Sl`

The `dimN` first columns of `X` span the maximal "complementary observability subspace" of `Sl`. (`dimS=0` if B(ker(D))=0).

If `flag='st'` is given, a five blocks partition of the matrices is returned and `dims` has four components. If `flag='pp'` is given a four blocks partition is returned (see abinv).

This function can be used to calculate an unknown input observer:

    // DDEP: dot(x)=A x + Bu + Gd
    //           y= Cx   (observation)
    //           z= Hx    (z=variable to be estimated, d=disturbance)
    //  Find: dot(w) = Fw + Ey + Ru such that
    //          zhat = Mw + Ny
    //           z-Hx goes to zero at infinity
    //  Solution exists iff Ker H contains Sg(A,C,G) inter KerC (assuming detectability)
    //i.e. H is such that:
    // For any W which makes a column compression of [Xp(1:dimSg,:);C]
    // with Xp=X' and [X,dims,J,Y,k,Z]=cainv(syslin('c',A,G,C));
    // [Xp(1:dimSg,:);C]*W = [0 | *] one has
    // H*W = [0 | *]  (with at least as many aero columns as above).

### See Also

`abinv`, `dt_ility`, `ui_observer`,